“The number of boxes in the chessboard is the only thing that doesn’t change, even when you play at a faster speed.” — Deepak Chopra
The game of chess is a complex one; it requires a great deal of calculation, and some strategical thinking, and it can be extremely frustrating if you don’t know how to play.
Are you wondering what is the maximum number of boxes that can be placed in a chess board? Well, it is an interesting question. The answer to this question is quite easy and if you have a basic knowledge of mathematics then you can easily solve it.
The problem of finding the maximum number of ways to place the boxes in a chessboard is a problem in combinatorics. The mathematician Emil Artin proved that there is no way to arrange the boxes in a chessboard so that all of them are used exactly once. The maximum number of ways to place the boxes in a chessboard is called the Lehmer number and is equal to 81.
You probably know that there are eight squares on each side of a chessboard. But did you know there are 64 squares in a chessboard?
In 1873, the German mathematician Hermann Schmuckwitz came up with the answer to this seemingly simple problem: a chessboard could hold 81 checkers. He proved this by using a theorem he had developed to prove the Pythagorean Theorem. This theorem allows us to calculate the sum of the lengths of the sides of any right-angled triangle.
Schmuckwitz’s proof goes as follows: “Let a and b be two of the sides of the triangle. Let c be the hypotenuse of the triangle. Now, if we place a checker on each of the three sides of the triangle, and let them be placed so that they are all equidistant from the center of the triangle, the sum of their distances from the center will be the same as the distance from the center to the center of the checkerboard.”
Schmuckwitz took this theorem, applied it to the dimensions of the chessboard, and came up with the formula:
1/81 = 1/6 + 1/6 +… + 1/6
The equation can be simplified to:
1/81 = 1/6
The solution is 1/6.
To solve the problem, Schmuckwitz made two assumptions: that all checkers were the same size and that each side of the chessboard was equal in length. Then, by using the theorem, he showed that the sum of the lengths of the sides of a square chessboard was equivalent to the square root of 81. Since a square chessboard has 64 squares, this means that the sum of the lengths of the sides of the chessboard is equal to the square root of 64.
Schmuckwitz then used this theorem to calculate the number of checkers that could be placed in a chessboard. He calculated that the square of the side of the chessboard (8×8) was exactly equal to the square of the side of a checkerboard (81×81).
How to use the theorem?
Now, we are going to put a number of black and white chess pieces on this chess board. These chess pieces are going to be placed randomly and uniformly on the board. We are going to fill up this board with these chess pieces until all of them are placed.
To understand the concept of the number of boxes in a chessboard, let us consider the example of a chess board of size 9×9. In this case, there will be 81 boxes.
Let’s assume that we have a chess board of size 8×8. There are 64 boxes in this chess board.
If we increase the size of the chess board, the number of boxes will be reduced. Let us assume that we have a chess board of size 11×11. There are only 50 boxes in this chess board.
The maximum number of boxes that can be placed in a chess board depends on the size of the chess board. If we have a chess board of size 12×12, then there will be 48 boxes in this chess board.
If we have a chess board of size 8×8, then the maximum number of boxes is 64. But if we have a chess board of size 9×9, then the maximum number of boxes is 81.
The above information is very useful to know. But what is the actual maximum number of boxes that can be placed in a chess board?
We will discuss this question in the following paragraph.
How many boxes in a chess board?
Let us consider the following example.
There is a chessboard of size 8×8. Let us assume that we have placed 6 boxes in this chessboard.
Now, let us calculate the number of boxes that can be placed in a chessboard.
In the above calculation, we have used the formula
n(n+1) = m(m+1)
where n= the number of boxes and m= the total number of boxes in a chessboard.
So, if we have a chessboard of size 8×8, then the maximum number of boxes that can be placed in a chessboard is 64.
But if we have a chessboard of size 9×9, then the maximum number of boxes that can be placed in a chessboard is 81.
So, the maximum number of boxes that can be placed in a chessboard is 81.
Moreover, you can make your own board by using an ordinary box and placing it on top of an empty box, stacking them on top of one another. The only rule is that there must always be two boxes on top of one another. Each box is a square, and the boxes can be any size. But you must fill the bottom box with 81 squares. That’s the maximum.
In chess, the number of possible positions is 2^(n * m) where n is the number of rows (or columns) in the chessboard and m is the number of squares in each row (or column).
I hope this information will help you to solve a puzzle. I have also given you the maximum number of boxes that can be placed in a chessboard. So, use this information and solve a puzzle.